Negative-energy contributions to transition amplitudes in heliumlike ions

Transcription

1 PHYSICAL REVIEW A VOLUME 58, NUMBER 6 DECEMBER 998 Negatve-energy contrbutons to transton ampltudes n helumlke ons A. Derevanko, Igor M. Savukov, and W. R. Johnson Department of Physcs, Notre Dame Unversty, Notre Dame, Indana D. R. Plante Department of Mathematcs & Computer Scence, Stetson Unversty, DeLand, Florda Receved 9 May 998 We derve the leadng term n an Z expanson for the negatve-energy vrtual electron-postron par contrbutons to the transton ampltudes of helumlke ons. The resultng expressons allow us to perform a general analyss of the negatve-energy contrbutons to electrc- and magnetc-multpole transton ampltudes. We observe a strong dependence on the choce of the zeroth-order Hamltonan, whch defnes the negatveenergy spectrum. We show that for transtons between states wth dfferent values of al spn, the negatveenergy contrbutons calculated n a Coulomb bass vansh n the leadng order whle they reman fnte n a Hartree bass. The rato of negatve-energy contrbutons to the al transton ampltudes for some of nonrelatvstcally forbdden transtons s shown to be of order /Z. In the partcular case of the magnetc-dpole transton 3 3 S 2 3 S, we demonstrate that the neglect of negatve-energy contrbutons, n an otherwse exact no-par calculaton, would lead one to underestmate the decay rate n helum by a factor of.5 n calculatons usng a Hartree bass and by a factor of 2.9 usng a Coulomb bass. Fnally, we tabulate revsed values of the lne strength S for the magnetc-quadrupole (M 2 ) transton 2 3 P 2 S 0. These values nclude negatve-energy contrbutons from hgher partal waves, whch were neglected n our prevous calculatons. S PACS numbers: 3.30.Jv, 3.5.Md, Cs I. INTRODUCTION The ab nto relatvstc consderaton of an atomc system requres a careful treatment of negatve-energy states vrtual electron-postron pars, whch, f ncluded mproperly, lead to the contnuum dssoluton problem dscussed by Sucher. An accepted remedy for ths problem s to use the no-par Hamltonan, whch excludes negatve-energy states,2. The no-par approach s well justfed for determnng energes of many-body systems. The leadng correctons beyond the no-par approxmaton have been studed for the ground state of helumlke ons n Ref. 3. However, the effect of negatve-energy states on the transton ampltudes stll remans an open queston. If such correctons were a neglgbly small fracton of the al ampltude, the study would be manly of academc nterest. However, as we demonstrate n ths work, the relatve contrbuton for some nonrelatvstcally forbdden transtons s of order /Z and thus the practcal mportance of understandng when to nclude negatve-energy contrbutons cannot be overstated. To allow general consderaton we wll derve the leadng term n the Z expanson of the negatve-energy contrbuton to the transton ampltudes n helumlke ons. The separaton of negatve- and postve-energy states depends on the choce of the zeroth-order Hamltonan. In practcal calculatons one often employs the Drac-Coulomb Hamltonan modfed by a model potental, chosen to approxmate the nteracton between electrons. The queston of negatve-energy contrbutons to the magnetc-dpole transton 2 3 S S 0 was consdered numercally by Lndroth and Salomonson 4 for helumlke argon and by Indelcato 5 n systematc multconfguratonal Hartree-Fock MCHF calculatons. Employng a Drac-Coulomb bass set, the former work demonstrated numercally that the negatveenergy contrbuton vanshes. By contrast, Indelcato s MCHF calculatons revealed contrbutons from negatveenergy states that were comparable to the regular postve-energy values, especally for lght ons. We wll address ths dfference by dervng negatve-energy correctons both for the Drac-Coulomb bass and for a bass modfed by a sphercally symmetrc model potental. The transton ampltudes, although calculated by employng exact egenfunctons of the no-par Hamltonan, depend on the gauge of the electromagnetc feld, as noted n Ref. 6. Ths gauge dependence s reflected n slght dfferences between the length-form and velocty-form transton ampltudes. In Ref. 7 t was demonstrated that the gauge dependence s a drect consequence of the omsson of contrbutons from negatve-energy states n the no-par Hamltonan. In that work the confguraton-nteracton CI method was used to determne the contrbuton to transton ampltudes from postve-energy states and perturbaton theory was used to obtan the contrbuton from the negatve-energy states. Based on numercal analyss, the authors of Ref. 7 noted that the negatve-energy contrbutons were neglgble for length-form transton ampltudes compared to those calculated n the velocty form. They recommended usng the length-form no-par ampltudes n calculatons of the electrc-dpole transton ampltudes. We extend ths recommendaton to all electrc-multpole transtons. For magnetc-multpole transton ampltudes, we fnd that when calculated n a Drac-Coulomb bass, the leadng term vanshes for transtons between states wth dfferent values of al spn due to the cancellaton between Coulomb and Bret contrbutons. Snce all magnetc-multpole transtons to the ground state of helumlke ons ( S 0 ) nvolve the /98/586/44539/$5.00 PRA The Amercan Physcal Socety

2 4454 DEREVIANKO, SAVUKOV, JOHNSON, AND PLANTE PRA 58 upper state wth al spn S, we make a general asserton that, for such M J transtons, the leadng contrbuton due to negatve-energy states vanshes. Ths fact allows us to correct the numercal results of Ref. 7 for the magnetcquadrupole (M 2 ) transton 2 3 P 2 S 0. We tabulate revsed values of the lne strength S for ths transton. The negatve-energy correctons are of partcular mportance n the case of nonrelatvstcally forbdden magnetc-dpole transtons. We consder the transton 3 3 S 2 3 S usng both Drac-Coulomb and Hartree bass sets. We fnd the rato of negatve- to postve-energy contrbutons to be of order /Z n both calculatons. Furthermore, we fnd that for ths transton the neglect of contrbutons from the negatveenergy states would underestmate the decay rate by a factor of.54 for helum n the Hartree case and by a factor of 2.9 n the Coulomb case. We also demonstrate that modfcaton of the zeroth-order Hamltonan by a model potental ntroduces an addtonal correcton that gves rse to the leadng order for magnetcmultpole transtons between states wth dfferent al spn. Thus, even for the well-studed magnetc-dpole transton 2 3 S S 0, the rato of negatve- to postve-energy state contrbutons s shown to scale as /Z. We derve the analytcal expressons n Sec. II. The dscusson of negatve-energy contrbutons to electrcmultpole transtons s gven n Sec. III and to magnetcmultpole transtons n Sec. IV. Predctons based on analytcal results are llustrated by drect summaton over negatve-energy states n second-order many-body perturbaton theory. The numercal methods employed are dscussed n Sec. V. Conclusons are gven n Sec. VI. II. THEORY We perform a perturbaton theoretc analyss of transton ampltudes smlar to that gven n Ref. 7. The summaton over negatve-energy states appears for the frst tme n the second-order expresson. Here we brefly recaptulate the dervaton of the second-order perturbaton expresson for transton ampltudes. The many-electron Hamltonan H s represented as a sum of a zeroth-order Hamltonan H 0 and a perturbaton V, H 0 h 0 h D U, 2. V j V, j U, 2.2 where h D s the Drac Hamltonan of an electron n the Coulomb potental of a nucleus, U s a model potental, and the two-electron nteracton V(, j) s a sum of Coulomb and Bret nteractons. The egenfunctons of h 0 serve as a bass for perturbaton theory. They nclude both negatve- and postve-energy states. The ncluson of the model potental U n the zeroth-order Hamltonan can mprove the convergence of perturbaton theory f the model potental approxmates the Coulomb nteracton between atomc electrons. For helumlke systems, we wll consder both the Coulomb case, where the model potental U s set to zero, and the Hartree case, wth the self-consstent Hartree potental of the ground state v 0 (s,r). In the no-par approach the perturbaton V s surrounded by postve-energy projecton operators, thus elmnatng contrbutons from negatve-energy states completely to all orders of perturbaton theory. However, even the process of separaton of postve- and negatveenergy states depends on the choce of model potental U and would be acceptable f the fnal result were modfed only slghtly. In contrast, we wll demonstrate that the contrbuton of negatve-energy states to transton ampltudes depends senstvely on the choce of the model potental, especally for nonrelatvstcally forbdden transtons. In the zeroth-order approxmaton, the uncoupled states are antsymmetrzed combnatons of products of egenfunctons of the Hamltonan h 0. In the second-quantzed form, the ntal state s represented as (0) va a v a a 0 and the fnal state as (0) wb a w a b 0. Orbtal labels a and b refer to the s state and w and v to excted states. The possblty of v beng as electron s also allowed n the followng. We treat the nteracton V as a perturbaton and obtan the frst-order correcton to wave functons. For example, the correcton to the ntal state s va jva a v jva j v a a a j 0 v U a a a v a 0. U v v a a a Here the summaton s performed over egenstates of h 0 wth energes and v jkl are matrx elements of the two-partcle nteracton V(, j) n that bass. We consder a matrx element of a one-body operator H I h I () between two states of a helumlke on. Wth the ad of the frst-order correcton to wave functons, we form the expresson for the second-order matrx element and obtan T 2 wb H I 0 va 0 wb H I va T 2 awb v wba h I v v wba h I v w vwb v wbv h I a v wbv h I a v w b bva h I w v bva h I w v bva v h I b v wva h I b v wva w v a w U w h I v ab U w h I a bv w 2.4 h I w U v ab h v v I w U a bv. 2.5 a a The summaton over ntermedate states ncludes both negatve- and postve-energy states. It s our am to estmate the leadng term n the Z expanson due to summaton over the negatve-energy states. In the second-order expressons

3 PRA 58 NEGATIVE-ENERGY CONTRIBUTIONS TO TRANSITION above we have omtted the dervatve term gven n Ref. 7 snce t does not affect the negatve-energy state contrbuton. When consderng transton between states represented n the lowest order by a combnaton of product wave functons, e.g., the 2 3 P state, the above expressons should be generalzed. Such gauge-ndependent multconfguratonal generalzaton for Be-lke systems can be found n Ref. 8. A. Summaton over negatve-energy states We approxmate the dfferences between postve and negatve energes n the denomnators of Eq. 2.5 by 2mc 2. In ths paper we wll use atomc unts (m e e/4 0,c/). The Paul approxmaton for a postron wave functon becomes pˆ 2 c. 2.6 Nonrelatvstc postron wave functons form a complete bass set and thus satsfy the closure relaton r,s r,s rr s,s, 2.7 where s s a spn varable. Ths relaton allows us to express a summaton over negatve-energy states of electrons n the Paul approxmaton as r,s r,s r,srr s,s Here the matrx operator s defned as pˆ 4 r,s 2 c 2 pˆ 2 c. 2.9 pˆ 2 c The operator, when acted upon a wave functon, decreases the sze of the large component by an order of (Z) 2, makng the large and small components of the result comparable. Employng the modfed closure relaton 2.8, we fnd that the entre contrbuton from negatve-energy states to the second-order matrx element 2.5 s T 2 2c 2 W wbvaw bwav W wbav W bwva, 2.0 where the operator W s WV,2h I h I V,2 Uh I h I U. 2. Here and below the arguments and 2 stand for coordnate and spn varables of electrons and 2, respectvely. The operator W can be represented n a form symmetrc wth respect to nterchange of electrons and 2; however, we fnd the representatons 2.0 and 2. more convenent. At ths pont, we have performed a summaton over negatveenergy states for an arbtrary one-body operator H I. Now we turn to the calculaton of the correcton from negatve-energy states to transton ampltudes. B. Negatve-energy correctons to transton ampltudes For a sngle electron, the nteracton Hamltonan wth an electromagnetc feld descrbed by the vector potental A and scalar potental s gven by h I c A. 2.2 The two-electron nteracton V s a sum of Coulomb C and Bret B nteractons C 2 r 2, B 2 2 r r We employ the Paul approxmaton n calculatons of matrx elements of the operator W, where ths approxmaton for the electron wave functon s gven by p, 2.5 2c beng the nonrelatvstc wave functon of the electron. Below we gve a breakdown of contrbutons to the operator W arsng from varous combnatons of nteractons WW C W B W U W CA W BA W UA. 2.6 The dervaton of these contrbutons s based on commutator denttes for Paul matrces. W C s a contrbuton arsng from Coulomb nteracton and scalar potental W C 2 2 r There s a smlar contrbuton from the model potental W U, W U 2 2 U. 2.8 (2) These terms contrbute to T correctons of order (Z) 3, where desgnates scalng of the scalar potental. The contrbuton from the Bret nteracton and scalar potental W B s (Z) 2 smaller than W C and does not contrbute to the leadng order. The stuaton s dfferent for the vector potental part of the nteracton Hamltonan: The contrbuton from the vector potental and Coulomb nteracton W CA s of the same order as the contrbuton from the vector potental and Bret nteracton W BA. We have W CA A 2.9 r 2,

4 4456 DEREVIANKO, SAVUKOV, JOHNSON, AND PLANTE PRA 58 W BA 2 A r 2 r 2 A pˆ2 r 2 n 2 An 2 pˆ2, 2.20 where n 2 s the unt vector along r 2 r r 2. Fnally, the model potental combned wth the vector potental term results n an effectve operator TABLE I. Scalng of the rato of negatve- to postve-energy states contrbutons for electrc-multpole E J transton ampltudes n helumlke ons. Gauge Bass S0 S0 length Coulomb 4 Z 3 4 Z 3 velocty Coulomb 2 Z 2 Z length model potental 4 Z 3 4 Z 3 velocty model potental 2 Z /Z W UA A U. 2.2 The contrbuton to T (2) due to the vector potental scales as (Z)c A, where we explctly separated scalng of c A. For further dscusson, t s mportant to note that n the absence of a model potental the entre negatve-energy contrbuton vanshes for transtons between states wth dfferent values of al spn S. Indeed, combnng Coulomb and Bret contrbutons for the vector potental and expressng the sum n terms of the al spn S, we obtan W (CB)A W CA W BA, W CBA 2S A r 2 r 2 A pˆ2 r 2 n 2 An 2 pˆ The last two terms n the above expresson do not depend on spn and vansh for transtons between states wth S0. The frst term s proportonal to the al spn S and also vanshes for such transtons snce the reduced matrx element S SS 2 S S 2. Thus the leadng negatve-energy contrbuton vanshes for transtons between states wth dfferent values of the al spn when calculatons are performed n a Coulomb bass. The model potental term W UA, however, does not possess ths property and results n sgnfcantly dfferent contrbutons to ampltudes for transtons wth S0, as dscussed n the followng sectons. The analyss of the above expressons allows us to make general qualtatve predctons about the role of negatve-energy states n calculatons of transton ampltudes. We performed angular reductons of the negatve-energy contrbutons and the resultng expressons can be found n the Appendx. Alternatvely, the drect numercal summaton n second-order expressons 2.5 over the negatve-energy part of bass set of h 0 can be done to obtan the results for model potental case and to evaluate the hgher-order correctons. III. ELECTRIC-MULTIPOLE TRANSITIONS For practcal purposes we consder the multpole expanson of the electromagnetc feld, dscussed, for example, n Ref. 9, and fnd the matrx elements of the operator wth Q JM, j q JM j a a j, q JM j 4J 2J!! 2JJ k J c JM r,c A JM r, j. 3. Here J s the multpolarty, wth for electrc and 0 for magnetc transtons. The partcular form of electromagnetc multpole potentals depends on the choce of gauge and we refer the reader to Ref. 7 for explct expressons. We wll consder two forms of electromagnetc potentals: length and velocty transverse forms. In the velocty form the electrc scalar potental s dentcally zero. In the length form the above expressons reproduce the electrc J- pole moment operator () n the long-wavelength approxmaton (kr). The transton ampltude must reman nvarant under gauge transformaton. However, the calculatons performed n the no-par approach lack ths mportant property because of omsson of negatve-energy states, as has been demonstrated n Ref. 7. In ths secton we determne the effect of negatve-energy states on calculatons of the reduced matrx elements of Q () J for electrc-multpole E J transtons. The scalng of the rato of negatve- to postve-energy contrbutons for electrc-multpole transtons n both length and velocty gauges s summarzed n Table I. In ths table the scalng of the postve-energy states contrbuton for the nonrelatvstcally forbdden ntercombnaton (S0) transtons s assumed to be (Z) 2 tmes smaller than that of spn-allowed transtons. Also, the order of the next term n the negatve-energy contrbutons s assumed to be a factor of (Z) 2 smaller than the order of the leadng term. It has been shown numercally n Ref. 7 that for electrcdpole transtons 2 3 P S and 2 3 P S, negatveenergy correctons to transton ampltudes calculated n the length gauge are much smaller than those obtaned n the velocty gauge. Wth the help of our analytcal result, ths observaton can be extended for all electrc-multpole E J transtons. Indeed, the rato of negatve-energy contrbutons n the length form to those obtaned n the velocty form s (Z) 2 or less. Qualtatvely, such a substantal dfference between negatve-energy state contrbuton n length and velocty forms can be understood from the fact that the electrcdpole operator n velocty form mxes large and small components of wave functons, whle the length-form operator does not. On the other hand, the large and small components of postron wave functon have the opposte meanng. Ths leads to the fact that the matrx element of dpole operator between negatve- and postve-energy states n the

5 PRA 58 NEGATIVE-ENERGY CONTRIBUTIONS TO TRANSITION TABLE II. Contrbutons from negatve-energy states to the reduced matrx element of the electrc-dpole moment for the 2 3 P S 0 transton n helumlke ons calculated wth the Hartree bass set. (Q ) np s the no-par contrbuton from the second-order perturbaton theory calculatons. (Q ) C,(Q ) B, and (Q ) U are the negatve-energy contrbutons due to Coulomb and Bret nteractons and the model potental, respectvely. The breakdown for both length and velocty forms s presented. The notaton ab desgnates a 0 b. Gauge Z (Q ) np (Q ) C (Q ) B (Q ) U (Q ) Length Velocty Length Velocty Length Velocty Length Velocty second-order expressons s (Z) 2 tmes smaller n the length form than n the velocty form. There s a very surprsng effect for ntercombnaton transton ampltudes calculated n the velocty form n a model potental bass: The relatve contrbuton of negatve-energy states scales as /Z. Ths fact mples a sgnfcant correcton for lght ons. The reason s that even though the contrbutons from Bret and Coulomb nteractons cancel each other owng to S0, the correcton from the model-potental term stll remans and contrbutes n leadng order. A smlar behavor s found for nonrelatvstcally forbdden magnetcdpole transtons dscussed n Sec. IV. As an example, we consder the 2 3 P S 0 electrcdpole ntercombnaton transton. The calculaton requres a generalzaton of the second-order expressons to a multconfguratonal case snce the 2 3 P state s represented n the lowest order as a combnaton of (2p /2 s /2 ) and (2p 3/2 s /2 ) j-j coupled states. We employed the approach of Ref. 8 wth an obvous reducton to the case of helum. The results of the drect numercal summaton over a bass set n the second-order expressons are presented n Table II. Frst we note that the negatve-energy contrbutons brng the results of calculatons n length and velocty form nto agreement. In contrast to relatvely tny correctons found n Ref. 7 for allowed transtons, these contrbutons are at the level of 5% n velocty form for Z2, due to a specfc choce of transton. We also note that for low Z, the negatve-energy contrbutons n velocty form from the Coulomb nteracton cancels that from the Bret nteracton, so that the al negatve-energy correcton arses from the model potental term, as dscussed earler. The negatveenergy contrbuton would vansh f a Drac-Coulomb bass set were used. The relatve contrbuton of negatve-energy states s amplfed n the velocty form and s substantally smaller n the length form, as dscussed earler. Therefore, n order to reduce the effect of negatve-energy states on hgh-precson no-par calculatons, the length form of multpole-electrc transton operator should be employed. Also, n relatvstc multconfguratonal Hartree-Fock calculatons smlar to Ref. 5 one should take addtonal care for negatve-energy contrbutons to ntercombnaton transton ampltudes calculated n the velocty form. IV. MAGNETIC-MULTIPOLE TRANSITIONS In ths secton we consder the negatve-energy correcton to the magnetc-multpole operator M J 2cQ J (0). For magnetc-multpole transtons, the velocty and length forms of the multpole potentals are dentcal, the scalar potental beng zero. The scalng of ratos of negatve-energy to postve-energy contrbutons for magnetc-multpole transtons s summarzed n Table III. In ths table we explctly separate the magnetc-dpole (M ) transton snce t s nonrelatvstcally forbdden. There are several surprsng effects, whch we wll further explore n ths secton. It s worth notng that all magnetc-multpole transtons to the ground state of He-lke ons have S0. Therefore, the leadng order of the negatve-energy correcton calculated n the Coulomb bass vanshes. In other words, for transtons wth S0 the contrbuton arsng from the Coulomb nteracton gven by Eq. 2.9 exactly cancels the contrbuton from the Bret nteracton As mentoned before, such detaled cancellaton for the magnetc-dpole transton 2 3 S S 0 has been observed n the numercal calculatons of Ref. 4 for helumlke Ar (Z8). To llustrate ths pont further, we perform a drect numercal summaton over a set of negatve-energy states n Eq We present the absolute values of the ratos of negatve-energy contrbutons to the al value of the reduced matrx element n Fg.. The al value of the reduced matrx element has been taken from Ref. 7. From Fg. t s clear that the cancellaton between the Coulomb and Bret negatve-energy contrbutons s nearly complete for low values of Z, wth the al negatve-energy contrbuton beng less than one- TABLE III. Scalng of the rato of negatve- to postve-energy state contrbutons for magnetc-multpole M J transton ampltudes n helumlke ons obtaned n the transverse gauge. Bass S0 S0 M, Coulomb /Z 2 Z M, model potental /Z /Z J, Coulomb 2 Z 4 Z 3 J, model potental 2 Z 2 Z

6 4458 DEREVIANKO, SAVUKOV, JOHNSON, AND PLANTE PRA 58 FIG.. Comparson of the relatve contrbutons of the negatveenergy states due to the Coulomb and Bret nteractons to the al value of the reduced matrx element M for the magnetc-dpole transton 2 3 S S 0 along the helum soelectronc sequence. All contrbutons are scaled to the al value of M from Ref. 7. The dotted lne represents the Coulomb contrbuton, the dashed lne, the Bret contrbuton, and the sold lne, the al relatve contrbuton from the negatve-energy states. thousandth the ndvdual Coulomb and Bret contrbutons for Z5. For larger values of Z, there s only a partal cancellaton because terms of the hgher powers of Z contrbute to the negatve-energy correcton. It s worth notng that f there were no such cancellaton, the relatve contrbuton of negatve-energy states would scale as /Z. However, when the calculatons of magnetc-multpole transton ampltudes for S0 are performed wth the model potental bass set, the term W UA gves a leadng negatve-energy contrbuton and must be taken nto account. Indeed, for nonrelatvstcally forbdden magnetc-dpole transtons, e.g., 2 3 S S 0, one expects the rato of negatve- to postve-energy contrbutons to be of order of /Z. In Table IV we gve a numercal breakdown for the 2 3 S S 0 transton for selected values of Z. Agan, we fnd a strong cancellaton between the Coulomb and Bret contrbutons at low Z, wth less cancellaton as we ncrease the value of Z. Unlke the case of startng from a Coulomb bass, however, startng from the Hartree bass provdes us wth an addtonal component that s not neglgble. In the partcular case of helum, the negatve-energy states n a Hartree bass contrbute % to the al transton ampltude and 2% to the al transton rate. By contrast, the negatveenergy contrbutons are entrely neglgble at least n the second order n a Coulomb bass. It s worth emphaszng that the no-par CI values are calculated wth a hgh degree of accuracy. For hgh Z the fnal al n the Table IV s ndependent of the choce of bass. The 0.3% dfference of al values for Z2 represents a lmtaton of the present second-order treatment of the negatve-energy contrbutons. Recently, the effect of negatve-energy contrbutons for the 2 3 S S 0 M transton n the case of multconfguratonal Hartree-Fock calculatons has been consdered by Indelcato 5. Hs results also exhbt a /Z-lke rato of negatve- to postve-energy contrbutons that can be attrbuted to a breakdown of detaled cancellaton between Coulomb and Bret terms by the MCHF effectve model potental. The leadng term of the negatve-energy contrbuton n a Coulomb bass should also vansh for the magnetcquadrupole transton (M 2 ) transton 2 3 P 2 S 0. Such a cancellaton s demonstrated numercally n Table V. In Table V we also make a comparson wth the data gven n Ref. 7. The contrbutons are dfferent by several orders of magntude for low-z ons. The error was traced to an nsuffcent number of partal waves employed n calculatng the negatve-energy contrbutons n Ref. 7. We revse the numercal calculatons and fnd agreement wth the analytcal predctons. Incorrect values of negatve-energy contrbutons were used n 7 to determne the transton rate (A coeffcent and the lne strength S. Beng small, the negatve-energy state contrbuton dd not change the values of A at the level of sgnfcant fgures quoted; however, the lne strength S was quoted to a hgher degree of accuracy and we fnd some dfference due to negatve-energy correctons. Revsed values of lne strength are presented n Table VI. The dfference between the current results and those of Ref. 7 s n the last one or two sgnfcant fgures and amounts to 0.02% for Z00 and less for other ons. There s an nterestng case n whch calculatons startng from ether a Coulomb bass or a model potental bass would lead to negatve-energy contrbutons that are comparable to those of the postve-energy states. Ths s the case of the magnetc-dpole transtons between states of the same value of al spn. TABLE IV. Contrbutons from negatve-energy states to the reduced matrx element of the magnetcdpole moment for the 2 3 S S 0 transton n helumlke ons calculated wth Hartree and Drac-Coulomb bass sets. (M ) CA and (M ) BA are the negatve-energy contrbutons due to the Coulomb and Bret nteractons, respectvely. (M ) UA s a contrbuton from the Hartree model potental. (M ) s the sum of negatve-energy contrbutons. (M ) np s the no-par contrbuton from CI calculatons. The notaton a b desgnates a0 b. Z2 Z50 Z00 Contrbutons Hartree Coulomb Hartree Coulomb Hartree Coulomb CA (M ) BA (M ) UA (M ) (M ) np (M ) (M )

7 PRA 58 NEGATIVE-ENERGY CONTRIBUTIONS TO TRANSITION TABLE V. Contrbutons from negatve-energy states to the reduced matrx element of the magnetc-quadrupole moment for the 2 3 P 2 S 0 transton n helumlke ons calculated wth the Coulomb bass set. (M 2 ) CA and (M 2 ) BA are the negatve-energy contrbutons due to Coulomb and Bret nteractons, respectvely, and (M 2 ) s ther sum. The last column gves the erroneous negatve-energy contrbuton of Ref. 7. The notaton ab desgnates a0 b. Z (M 2 ) CA (M 2 ) BA (M 2 ) (M 2 ) a a Reference 7. TABLE VI. Revsed values of lne strength S n a.u. for the magnetc-quadrupole (M 2 ) transton 2 3 P 2 S 0 n helumlke ons. The notaton ab desgnates a0 b TABLE VII. Breakdown of contrbutons to the reduced matrx element of the magnetc-dpole moment and the correspondng transton rate for the 3 3 S 2 3 S transton n helumlke ons calculated wth Hartree bass sets. (M ) np s the no-par contrbuton from CI calculatons, (M ) s the contrbuton of negatve-energy states, and (M ) s ther sum. The notaton ab desgnates a 0 b. Z (M ) np (M ) (M ) A (s ) For nonrelatvstcally forbdden magnetc-dpole transtons between states of the same value of al spn, the leadng term n the Z expanson of the postve-energy contrbuton to the transton ampltude vanshes, reducng the transton ampltude by a factor of (Z) 2. Thus one could expect the rato of negatve-energy contrbuton to the al transton ampltude be of order of /Z n ether the Coulomb or model potental bass. We consder the partcular magnetc-dpole transton 3 3 S 2 3 S. It s straghtforward to check that, due to angular selecton rules, the spnndependent terms n Eq vansh. Thus the negatveenergy contrbutons from the Bret and Coulomb nteractons are equal n leadng order. The calculatons based on Eq. 2.9 show that both contrbute the value of.70 6 Z a.u. to the transton ampltude. The analytcal value for the reduced matrx element for helum, obtaned wth the Eq. A4, s a.u. Ths value closely agrees wth the numercal value obtaned by a drect summaton over the negatve spectrum a.u. usng a Coulomb bass set. The numercal results for the reduced matrx element of magnetc-dpole moment and the correspondng transton rate for the 3 3 S 2 3 S transton are presented n the Table VII. The calculatons were performed n a Hartree model potental. The no-par contrbutons and the energes used for tabulaton of transton rates were obtaned wth relatvstc confguraton-nteracton method 0,. The Lamb shft was estmated usng screened Coulomb feld values followng tabulatons 2. The ncluson of the Lamb shft modfes the thrd sgnfcant fgure of the rate values for Z50. The negatve-energy contrbuton to the matrx element was calculated by a drect numercal summaton over negatve-energy states n the second-order expressons 2.5. The numercal detals of the calculatons are descrbed n Sec. V. The relatve negatve-energy contrbuton to transton ampltude s 20% for neutral helum and becomes smaller for larger Z, modfyng the ampltude by 0.3% for Z00. We also perform a smlar analyss for the Coulomb bass. For helum, the sum of the CI no-par ( a.u. and second-order contrbuton from negatve-energy states ( a.u. amounts to a.u. Ths value s almost by a factor of 3 larger than the correspondng value from calculatons n the Hartree bass. For hgh-z ons such a comparson becomes much better, e.g., for Z00; the calculatons n the Coulomb and Hartree bases agree to four sgnfcant fgures. We emphasze agan that the no-par CI values presented here are converged to a hgh accuracy. The dfferences between the resultng al ampltudes are due to the lmtaton of the second-order perturbaton theory treatment of the negatve-energy states. In other words, brngng the al values of the reduced matrx element calculated wth dfferent startng potentals nto agreement wth one another requres consderaton of negatve-energy contrbutons

8 4460 DEREVIANKO, SAVUKOV, JOHNSON, AND PLANTE PRA 58 FIG. 2. Comparson of the relatve contrbutons of the negatveenergy states to the transton rates (A coeffcents for varous transtons along the helum soelectronc sequence. The rato of A A no par /A s plotted., M 3 3 S 2 3 S, Coulomb bass set; *, M 3 3 S 2 3 S, Hartree bass set;, E 3 3 P S 0, velocty form, Hartree bass set;, M 2 3 S S 0, Hartree bass set;, M 2 3 S S 0, Coulomb bass set. n hgher orders of perturbaton theory. To estmate theoretcal uncertanty of the results calculated n the Hartree bass, we compare the second-order nopar calculatons wth the all-order no-par CI result. For helum, the second-order result ( a.u. recovers 96% of the all-order value ( a.u., wth better agreement for hgher Z. Snce the negatve-energy states contrbute at the level of 20% for He, we estmate the theoretcal uncertanty of the al value of the reduced matrx element to be at % level. Such relatve theoretcal error n the al value of transton ampltude scales as /Z 2 for hgher Z. The relatve modfcaton of the al rate by the ncluson of negatve-energy states for 3 3 S 2 3 S transton s shown n Fg. 2. Both Coulomb and Hartree cases are consdered. The no-par values for both cases are taken from CI calculatons. The al rate s obtaned from the reduced matrx element values presented n Table VII. The neglect of contrbutons from negatve-energy states would underestmate the decay rate n the Hartree case by a factor of.54 for neutral helum and underestmate the rate by 0.6% for Z 00. The rates calculated n the Coulomb bass are affected more strongly and the omsson of negatve-energy states would underestmate the decay rate n the Hartree case by a factor of 2.9 for neutral helum and underestmate the rate by 0.8% for Z00. V. NUMERICAL DETAILS For our numercal calculatons, the B-splne bass set has been employed for both the CI and many-body perturbaton theory MBPT calculatons. The B-splne bass functons approxmate the egenfunctons of h 0 n a sphercal cavty. The reader s referred to Ref. 3 for detals. For the transtons consdered prevously n Ref. 7, the same bass sets and cavty rad are used. As noted prevously, an nsuffcent number of partal waves were employed n the negatveenergy calculatons of Ref. 7 for the 2 3 P 2 S 0 magnetc-quadrupole transton. The addton of these partal waves marks the only change n the present calculatons for ths transton from those of Ref. 7. To our knowledge, the magnetc-dpole 3 3 S 2 3 S transton has not been dscussed prevously n the lterature and we present the calculatons n detals. For ths transton, convergence wth respect to bass set sze, cavty radus, and number of partal waves ncluded was not as straghtforward as for the other transtons. Our am was to obtan an accuracy of a part n 0 5. Because of the ncluson of n3 states nto the lowest-order wave functon, the cavty radus was ncreased to 20/Z a.u. n sze. Unlke the other transtons, however, 40 splnes were nsuffcent n convergng our solutons for low Z. Even when a 50 splne bass was used, the second-order MBPT values dffered by as much a % when varyng the cavty radus between 40 and 60 a.u. for the case of neutral He. We therefore used an th-order, 70-splne bass set to obtan both our MBPT and CI results for Z 2 5. Whle such a large bass was requred however, we only needed to sum over 30 of the 70 splnes to obtan the desred level of accuracy. For Z0, only a 50-splne bass was requred, summng over the lowest 25 splnes. For Z 20, a 40-splne bass was used, agan summng only over the lowest 25 splnes. The relatvstc confguraton-nteracton method, employed for calculatons n ths paper, has been descrbed n Refs. 0,. The expresson for matrx elements n the CI framework s gven n Ref. 7. For the CI calculatons of the 3 3 S 2 3 S transtons, the bass orbtals were lmted to those havng orbtal angular momentum l4 for neutral He. For all other Z50, ncludng only orbtals wth l3 was suffcent. For Z50, only orbtals wth l2 were requred. VI. CONCLUSIONS We have dscussed the role of negatve-energy states n calculatons of transton ampltudes for helumlke ons by dervng the leadng term of ther contrbuton. The scalng of ratos of negatve- to postve-energy state contrbutons to transton ampltudes s presented n Table I for electrcmultpole transtons and n Table III for magnetc-multpole transtons. We determned several stuatons where the contrbutons of negatve-energy states are comparable to the al value of transton ampltude and must be taken nto account. In contrast to the usual understandng of relatvstc effects, the negatve-energy states contrbutons for the followng cases are most mportant for low-z calculatons: electrc-multpole (E J ) ntercombnaton transton ampltudes calculated n velocty form n a model potental bass, magnetc-dpole (M ) transton ampltudes calculated n a model potental bass between states wth dfferent values of al spn, and magnetc-dpole (M ) transton ampltudes calculated n any bass between states wth the same value of al spn. Fgure 2 demonstrates how the ncluson of negatveenergy states n the calculatons modfes the transton rate for the enumerated cases. It s clear that the negatve-energy states play a sgnfcant role n the determnaton of the al rate, contrbutng for low Z at the level of a few percent for the magnetc-dpole 2 3 S S 0 transton Hartree bass, 0% for the velocty form of the electrc-dpole ntercombnaton 2 3 P S 0 transton Hartree bass, and 00% for the magnetc-dpole 3 3 S 2 3 S transton both Coulomb and Hartree sets. The nonsmooth behavor of the curve for the magnetc-dpole 2 3 S S 0 transton Hartree ba-

9 PRA 58 NEGATIVE-ENERGY CONTRIBUTIONS TO TRANSITION ss s explaned by the change of sgn of the negatve-energy contrbuton at Z23. The negatve-energy contrbuton to transton ampltudes calculated n a Coulomb bass s smaller than or comparable to that calculated n a model potental bass. In order to decrease the effect of negatve-energy states n electrcmultpole transtons, one should employ the length form for electrc-multpole potentals, as noted n Ref. 7. Fnally, we have tabulated revsed values for lne strengths of magnetc-quadrupole 2 3 P 2 S 0 transtons n helumlke ons. These values were obtaned from the relatvstc no-par confguraton-nteracton calculatons wth negatve-energy contrbutons added from the second-order perturbaton theory. ACKNOWLEDGMENTS A.D. and W.R.J. were supported n part by NSF Grant No. PHY The authors owe thanks to H.G. Berry for useful dscussons and comments on the manuscrpt. APPENDIX: ANGULAR REDUCTION OF THE NEGATIVE-ENERGY CONTRIBUTION TO MAGNETIC-DIPOLE MATRIX ELEMENT In ths secton we present the results of angular reducton for the leadng order n an Z expanson for matrx element of magnetc dpole. The startng expressons are gven by Eqs for the operator W. The results below are expressed n terms of radal ntegrals Rjkl 0 P r P k r r d dr 0 P j r 2 r P l r 2 dr 2dr. A Here P(r) s the radal part of the nonrelatvstc wave functon of the zeroth-order Hamltonan and r max(r,r 2 ). Also 0 s the zeroth-order transton energy and s the frst-order correcton to the transton energy. The analytcal expressons for 0 and can be found, for example, n Ref. 7. For the 2 3 S S 0 magnetc-dpole transton the negatve-energy contrbuton due to the Coulomb FM I CA and Bret FM I BA nteractons cancel each other, as dscussed n Sec. IV. The contrbuton due to the Coulomb nteracton s gven by FM I CA Rs,2s,s,s. R2s,s,s,s A2 If the zeroth-order Hamltonan ncludes a sphercally symmetrc model potental U, there s also a contrbuton FM I UA. The al negatve-energy contrbuton s represented by ths term. In the partcular case of the selfconsstent Hartree potental we obtan Urv 0 s,r 2 Ps r dr 0 r FMI FMI UA R2s,s,s,s. A3 In the case of the magnetc-dpole transton 3 3 S 2 3 S, the contrbutons from Coulomb and Bret nteractons are equal and FMI CA R3s,s,2s,s Rs,3s,2s,sR3s,s,s,2s Rs,3s,s,2s. A4 J. Sucher, Phys. Rev. A 22, M.H. Mttleman, Phys. Rev. A 4, ; 5, ; 24, S.A. Blundell, P.J. Mohr, W.R. Johnson, and J. Saprsten, Phys. Rev. A 48, ; I. Lndgren, H. Persson, S. Salomonson, and L. Labzowsky, bd. 5, Eva Lndrot and Sten Salomonson, Phys. Rev. A 4, P. Indelcato, Phys. Rev. Lett. 77, J. Hller, J. Sucher, G. Fenberg, and B. Lynn, Ann. Phys. N.Y. 27, W.R. Johnson, D.R. Plante, and J. Saprsten, Adv. At., Mol., Opt. Phys. 35, W.R. Johnson, U.I. Safronova, and A. Derevanko unpublshed. 9 A.I. Akhezer and V.B. Berestetsk, Quantum Electrodynamcs Interscence, New York, M.H. Chen, K.T. Cheng, and W.R. Johnson, Phys. Rev. A 47, K.T. Cheng, M.H. Chen, W.R. Johnson, and J. Saprsten, Phys. Rev. A 50, W.R. Johnson and Gerhard Soff, At. Data Nucl. Data Tables 33, W.R. Johnson, S.A. Blundell, and J. Saprsten, Phys. Rev. A 37,